lean/OAI/Combinatorics/IndependentSets/Encoding/Target.lean

import Mathlib

namespace OAI

namespace IndependentSetsGames.Foundations.Target

structure Literal («variables» : Nat) where
  variableIndex : Fin «variables»
  positive : Bool
  deriving DecidableEq

def Literal.eval {n : Nat} (literal : Literal n) (assignment : Fin n → Bool) : Bool :=
  if literal.positive then assignment literal.variableIndex else !(assignment literal.variableIndex)

abbrev Clause («variables» : Nat) := Vector (Literal «variables») 3

def Clause.eval {n : Nat} (clause : Clause n) (assignment : Fin n → Bool) : Bool :=
  (clause[0].eval assignment || clause[1].eval assignment) || clause[2].eval assignment

structure Formula where
  «variables» : Nat
  clauses : List (Clause «variables»)

def Formula.Satisfiable (formula : Formula) : Prop :=
  ∃ assignment : Fin formula.«variables» → Bool,
    ∀ clause ∈ formula.clauses, clause.eval assignment = true

structure PermutationTable (alphabet : Nat) where
  images : Vector (Fin alphabet) alphabet
  inverseImages : Vector (Fin alphabet) alphabet
  leftInverse : ∀ label : Fin alphabet, inverseImages[images[label]] = label
  rightInverse : ∀ label : Fin alphabet, images[inverseImages[label]] = label

theorem PermutationTable.images_injective {q : Nat} (table : PermutationTable q)
    {x y : Fin q} (h : table.images[x] = table.images[y]) : x = y := by
  have inverseEquality := congrArg (fun label : Fin q => table.inverseImages[label]) h
  simpa only [table.leftInverse] using inverseEquality

theorem PermutationTable.images_surjective {q : Nat} (table : PermutationTable q)
    (label : Fin q) : ∃ preimage : Fin q, table.images[preimage] = label :=
  ⟨table.inverseImages[label], table.rightInverse label⟩

structure Constraint (vertices alphabet : Nat) where
  source : Fin vertices
  target : Fin vertices
  permutation : PermutationTable alphabet

def Constraint.satisfied {n q : Nat} (constraint : Constraint n q)
    (labeling : Fin n → Fin q) : Bool :=
  decide (constraint.permutation.images[labeling constraint.source] = labeling constraint.target)

structure Instance (alphabet : Nat) where
  vertices : Nat
  constraints : List (Constraint vertices alphabet)
  nonempty : constraints ≠ []

def countSatisfied {n q : Nat} (labeling : Fin n → Fin q) :
    List (Constraint n q) → Nat
  | [] => 0
  | constraint :: rest =>
      (if constraint.satisfied labeling then 1 else 0) + countSatisfied labeling rest

theorem countSatisfied_le_length {n q : Nat} (labeling : Fin n → Fin q)
    (constraints : List (Constraint n q)) :
    countSatisfied labeling constraints ≤ constraints.length := by
  induction constraints with
  | nil => simp [countSatisfied]
  | cons constraint rest ih =>
      simp only [countSatisfied, List.length_cons]
      split <;> omega

theorem Instance.constraintCount_positive {q : Nat} (game : Instance q) :
    0 < game.constraints.length := by
  cases h : game.constraints with
  | nil => exact False.elim (game.nonempty h)
  | cons constraint rest => simp

structure RationalError where
  numerator : Nat
  denominator : Nat
  numeratorPositive : 0 < numerator
  denominatorPositive : 0 < denominator
  belowHalf : 2 * numerator < denominator

def CompleteAt {q : Nat} (error : RationalError) (game : Instance q) : Prop :=
  ∃ labeling : Fin game.vertices → Fin q,
    error.denominator * game.constraints.length ≤
      error.denominator * countSatisfied labeling game.constraints +
      error.numerator * game.constraints.length

def SoundAt {q : Nat} (error : RationalError) (game : Instance q) : Prop :=
  ∀ labeling : Fin game.vertices → Fin q,
    error.denominator * countSatisfied labeling game.constraints ≤
      error.numerator * game.constraints.length

structure SemanticGapReduction (completenessError soundnessError : RationalError) where
  alphabet : Nat
  alphabetAtLeastTwo : 2 ≤ alphabet
  reduce : Formula → Instance alphabet
  completeness : ∀ formula : Formula,
    formula.Satisfiable → CompleteAt completenessError (reduce formula)
  soundness : ∀ formula : Formula,
    ¬ formula.Satisfiable → SoundAt soundnessError (reduce formula)

end IndependentSetsGames.Foundations.Target

namespace IndependentSetsGames.Foundations.Complexity

def encodeWord (n : Nat) : List Bool := List.replicate n true ++ [false]

def encodeWords : List Nat → List Bool
  | [] => []
  | n :: ns => encodeWord n ++ encodeWords ns

def decodeWordsAux : List Bool → Nat → Option (List Nat)
  | [], 0 => some []
  | [], _ + 1 => none
  | true :: bs, n => decodeWordsAux bs (n + 1)
  | false :: bs, n => (decodeWordsAux bs 0).map (n :: ·)

def decodeWords (bs : List Bool) : Option (List Nat) := decodeWordsAux bs 0

theorem decodeWordsAux_word (n k : Nat) (bs : List Bool) :
    decodeWordsAux (encodeWord n ++ bs) k =
      (decodeWordsAux bs 0).map ((k + n) :: ·) := by
  induction n generalizing k with
  | zero => simp [encodeWord, decodeWordsAux]
  | succ n ih =>
      simp only [encodeWord, List.replicate_succ, List.cons_append, List.append_assoc,
        decodeWordsAux]
      simpa [encodeWord, Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using ih (k + 1)

@[simp] theorem decodeWords_encodeWords (ns : List Nat) :
    decodeWords (encodeWords ns) = some ns := by
  induction ns with
  | nil => rfl
  | cons n ns ih =>
      simp only [decodeWords, encodeWords, decodeWordsAux_word, Nat.zero_add]
      simpa [decodeWords] using congrArg (Option.map (n :: ·)) ih

theorem encodeWords_injective {xs ys : List Nat} (h : encodeWords xs = encodeWords ys) :
    xs = ys := by
  have decoded := congrArg decodeWords h
  simpa only [decodeWords_encodeWords, Option.some.injEq] using decoded

@[simp] theorem encodeWord_length (n : Nat) : (encodeWord n).length = n + 1 := by
  simp [encodeWord]

@[simp] theorem encodeWords_append (xs ys : List Nat) :
    encodeWords (xs ++ ys) = encodeWords xs ++ encodeWords ys := by
  induction xs with
  | nil => rfl
  | cons x xs ih => simp [encodeWords, ih, List.append_assoc]

@[simp] theorem encodeWords_length (ns : List Nat) :
    (encodeWords ns).length = ns.sum + ns.length := by
  induction ns with
  | nil => rfl
  | cons n ns ih => simp [encodeWords, ih]; omega

theorem encodeWords_length_le (ns : List Nat) (bound : Nat)
    (bounded : ∀ n ∈ ns, n ≤ bound) :
    (encodeWords ns).length ≤ ns.length * (bound + 1) := by
  induction ns with
  | nil => simp [encodeWords]
  | cons n ns ih =>
      have hn := bounded n (by simp)
      have hns := ih (fun x hx => bounded x (by simp [hx]))
      simp only [encodeWords, List.length_append, encodeWord_length, List.length_cons,
        Nat.add_mul, Nat.one_mul]
      omega

open Target

def literalWords {n : Nat} (literal : Literal n) : List Nat :=
  [literal.variableIndex.val, if literal.positive then 1 else 0]

def clauseWords {n : Nat} (clause : Clause n) : List Nat :=
  literalWords clause[0] ++ literalWords clause[1] ++ literalWords clause[2]

def formulaWords (formula : Formula) : List Nat :=
  [formula.«variables», formula.clauses.length] ++ formula.clauses.flatMap clauseWords

def formulaBits (formula : Formula) : List Bool := encodeWords (formulaWords formula)

@[simp] theorem literalWords_length {n : Nat} (literal : Literal n) :
    (literalWords literal).length = 2 := by simp [literalWords]

@[simp] theorem clauseWords_length {n : Nat} (clause : Clause n) :
    (clauseWords clause).length = 6 := by simp [clauseWords]

theorem clausesWords_length {n : Nat} (clauses : List (Clause n)) :
    (clauses.flatMap clauseWords).length = 6 * clauses.length := by
  induction clauses with
  | nil => rfl
  | cons clause clauses ih =>
      simp only [List.flatMap_cons, List.length_append, clauseWords_length,
        List.length_cons, ih, Nat.mul_add, Nat.mul_one]
      omega

@[simp] theorem formulaWords_length (formula : Formula) :
    (formulaWords formula).length = 2 + 6 * formula.clauses.length := by
  simp only [formulaWords, List.length_append, List.length_cons, List.length_nil,
    clausesWords_length]

theorem literalBits_length_le {n : Nat} (literal : Literal n) :
    (encodeWords (literalWords literal)).length ≤ n + 2 := by
  have hindex := literal.variableIndex.isLt
  cases hp : literal.positive <;> simp [literalWords, hp]

theorem clauseBits_length_le {n : Nat} (clause : Clause n) :
    (encodeWords (clauseWords clause)).length ≤ 3 * (n + 2) := by
  have h0 := literalBits_length_le clause[0]
  have h1 := literalBits_length_le clause[1]
  have h2 := literalBits_length_le clause[2]
  simp only [clauseWords, encodeWords_append, List.length_append]
  omega

theorem clausesBits_length_le {n : Nat} (clauses : List (Clause n)) :
    (encodeWords (clauses.flatMap clauseWords)).length ≤ clauses.length * (3 * (n + 2)) := by
  induction clauses with
  | nil => simp [encodeWords]
  | cons clause clauses ih =>
      have hc := clauseBits_length_le clause
      simp only [List.flatMap_cons, encodeWords_append, List.length_append,
        List.length_cons, Nat.add_mul, Nat.one_mul]
      omega

theorem formulaBits_length_le (formula : Formula) :
    (formulaBits formula).length ≤
      formula.«variables» + formula.clauses.length + 2 +
        formula.clauses.length * (3 * (formula.«variables» + 2)) := by
  have h := clausesBits_length_le formula.clauses
  simp only [formulaBits, formulaWords, encodeWords_append, List.length_append,
    encodeWords, encodeWord_length, List.length_nil]
  omega

end IndependentSetsGames.Foundations.Complexity

end OAI